I am a graduate student of Mathematics and currently studying measure theory.There is a lemma which is used to prove the Lebesgue dominated convergence theorem called Fatou's lemma.It states that:
If $(X,\mathcal S,\mu)$ is a measure space and $f_n:X\to [0,\infty]$ be measurable functions,then $\int_X \liminf\limits_{n \to \infty} f_n d\mu\leq \liminf\limits_{n\to \infty}\int_Xf_nd\mu$.
I am looking for some motivation behind this theorem.One motivation may be this:
We know that $\liminf$ is finitely super-additive i.e. $\liminf a_n+\liminf b_n\leq \liminf(a_n+b_n)$ for two sequences $(a_n)$ and $(b_n)$ in $\overline{\mathbb R}$.Now this can be generalized to $\liminf a_{1,n}+...+\liminf a_{N,n}\leq \liminf (a_{1,n}+...+a_{N,n})$ for $N$ number of sequences $(a_{1,n}),...,(a_{N,n})$ in $\overline{\mathbb R}$.Now define a sequence of functions $f_n:\{1,2,...,N\}\to \overline{\mathbb R}$ by, $f_n(k)=a_{k,n}$ then the above can be rewritten as $\sum\limits_{k=1}^N\liminf f_n(k)\leq \liminf\sum\limits_{k=1}^N f_n(k)$.So,it is natural to ask whether it is still valid if the sum is on a countably infinite index set say $\mathbb N$.Our lemma implies that if $X=\mathbb N$ and $\mathcal S=\mathcal P(\mathbb N)$ and $\mu$ is the counting measure,then for any sequence of non-negative functions $f_n:\mathbb N\to [0,\infty]$ ,which is measurable as $\mathcal S=\mathcal P(\mathbb N)$ we can write $\int_{\mathbb N}\liminf f_nd\mu\leq \liminf\int_{\mathbb N} f_nd\mu$ which is nothing but summation.So,we have, $\sum\limits_{k=1}^\infty \liminf f_n(k)\leq \liminf\sum\limits_{k=1}^\infty f_n(k)$.So,our initial result can be generalized to a countably infinite index set if the sequence is non-negative.
I like to think in this way.Is my intuition correct?